Method and apparatus for tracking a hand-held writing instrument with multiple sensors that are calibrated by placing the writing instrument in predetermined positions with respect to the writing surface

ABSTRACT

One embodiment of the writing instrument described herein comprises a pen-like device containing three accelerometers and three gyroscopes. Data from these sensors are processed via an Euler transform. Prior to using the writing instrument, the user holds the writing instrument in multiple predefined positions and sensor readings are taken. The sensor readings are mapped to the corresponding predefined positions. In addition, the sensor readings are compared to expected sensor readings to compensate for environmental variations such as altitude or temperature as well as for the possible tilt of a writing surface. The sensor readings are interpolated to map sensor readings other than those at the predefined positions to other writing instrument positions. When the user writes with the writing instrument, sensor outputs are used to track the writing instrument to define strokes made by the user. The data describing the strokes may be stored or processed to accomplish a variety of tasks including faxing a message, recording information, etc.

FIELD OF THE INVENTION

The present invention relates to tracking the position of a writinginstrument, and more specifically, to calibrating a system that tracksthe position of writing instrument.

BACKGROUND OF THE INVENTION

In the prior art, a number of input devices exist that allow a user toprovide input to a computer or similar device. Typically, these inputdevices (i.e., a mouse, a touch sensitive screen or pad) are physicallyconnected to the computer and do not allow a user to simply write theinput as he or she would with a writing instrument such as a pen or apencil. For some activities, such as sending a facsimile (faxing) ormaking an entry in an appointment book, it would be simpler for a userto write out a message or entry as he or she would with a writinginstrument that is not physically connected to the computer rather thanuse prior art input devices to create a message for faxing or to enteran appointment.

In order to provide the user with a writing instrument as an inputdevice, a system for tracking position(s) of a writing instrument mustbe provided. This system monitors movements of the writing instrumentand converts these movements into a representation of the user input.The better the tracking system, the more accurately the user input isreceived. However, because the writing instrument may be used in manydifferent environments where conditions such as altitude and temperaturevary, the tracking system to track the position of the writinginstrument may not perform accurately, resulting in the receipt ofincorrect input. Therefore, it is desirable to calibrate the writinginstrument for the environment in which the writing instrument is used.

SUMMARY OF THE INVENTION

A method and apparatus for calibrating a system that tracks the positionof a writing instrument is disclosed. Multiple sensors generate outputsbased on a position of the writing instrument. Sensor readings areobtained for multiple predefined positions of the writing instrument. Amapping of sensor outputs to writing instrument positions is generatedbased on the sensor readings at the predefined positions.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention is illustrated by way of example, and not by wayof limitation, in the figures of the accompanying drawings and in whichlike reference numerals refer to similar elements and in which:

FIG. 1 is one embodiment of a writing instrument tracked by a trackingsystem that may be calibrated according to the present invention.

FIG. 2 is a perspective view of one embodiment of a writing instrumenthaving three calibrating surfaces according to the present invention.

FIG. 3 is a perspective view of one embodiment of a writing instrumenthaving four calibrating surfaces according to the present invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

A method and apparatus for calibrating a system that tracks the positionof an input device such as a writing instrument is described. In thefollowing description, for the purposes of explanation, numerousspecific details are set forth in order to provide a thoroughunderstanding of the present invention. It will be apparent, however, toone skilled in the art that the present invention may be practicedwithout these specific details. In other instances, well-knownstructures and devices are shown in block diagram form in order to avoidobscuring the present invention.

The tracking system described herein may be used, for example, to trackthe position of a writing instrument such as a pen-based faxing devicedisclosed in a patent application entitled “METHOD AND APPARATUS FORPEN-BASED FAXING,” filed Sep. 30, 1997, application Ser. No. 08/940,832,or a data-entry device, such as disclosed in a patent applicationentitled “MANUAL ENTRY INTERACTIVE PAPER AND ELECTRONIC DOCUMENTHANDLING AND PROCESSING SYSTEM,” filed Nov. 1, 1995, application Ser.No. 08/551,535, both of which are assigned to the corporate assignee ofthe present invention. Such writing instruments allow a user to write ashe or she normally would write and the strokes made by the user are sentas a fax in the case of pen-based faxing or the strokes may be used fordata entry in the case of the data entry device.

Of course, other uses for the position tracking system of the presentinvention are also possible. For example, a wireless pointing devicesuch as the device disclosed in a patent application entitled “DIRECTPOINTING APPARATUS FOR DIGITAL DISPLAYS,” application Ser. No.08/840,552, filed Apr. 18, 1997, assigned to the corporate assignee ofthe present invention, may have a tracking system calibrated accordingto the present invention. Such a pointing device is not used for writingmessages, but instead performs precise tracking to accurately place acursor on a display.

In order to provide a writing device as described above, a method andapparatus is provided to calibrate the system that tracks position(s) ofthe writing instrument. In order to simplify the description of thepresent invention, a writing instrument is described. However, as to thedescription of the sensors and calibration of the system, thedescription applies equally to other embodiments, such as a cursorcontrol device.

One embodiment of the writing instrument described herein comprises apen-like device containing three acceleration sensors (accelerometers)and three angular velocity sensors (gyroscopes). Data from these sensorsare processed via an Euler transform. Prior to using the writinginstrument, the user holds the writing instrument in multiple predefinedpositions. Sensor readings are taken and mapped to the correspondingpredefined positions. In other words, sensor outputs (e.g., voltagelevels) are defined as the predefined positions by the tracking system.In addition, the sensor readings are compared to expected sensorreadings to compensate for environmental variations such as altitude ortemperature as well as for the possible tilt of a writing surface. Aftercalibration, sensor readings other than those at the predefinedpositions are mapped to corresponding positions by interpolating thesensor outputs obtained at the predefined positions. The mapping ofsensor outputs allows tracking of the position of the writing instrumentfor positions other than the predefined positions.

When the user writes with the writing instrument, sensor outputs areused to track the position of the writing instrument to define strokesmade by the user. The data describing the strokes may be stored orprocessed to accomplish a variety of tasks including those discussedabove, such as faxing a message, recording information, such as, forexample, an appointment or other tasks. In one embodiment, the strokedata comprises graphical representations of the strokes actually madewith the writing instrument rather than stored characters or templates,as with an optical character recognition (OCR) or similar device.Processing of the stroke data may occur within the writing instrument orat a remote computing device.

The Writing Instrument

One embodiment of a writing instrument having multiple sensors accordingto the present invention is briefly described. It should be noted,however, that the writing instrument is not required to make a physicalmark of any kind. Any device that allows a user to make writing motionsthat are to be tracked may be used. Additionally, because the actualstrokes that the user makes are tracked and processed, the quality ofthe user's handwriting is not important.

FIG. 1 is one embodiment of writing instrument 10 whose position may betracked according to the present invention. The lower portion of writinginstrument 10 generally comprises, ink supply 110, and pen tip 180. Asnoted above, writing instrument 10 is not required to actually mark asurface such as writing surface 190. Thus, ink supply 110 is onlyincluded when a pen-type writing instrument is required or discussed.Writing surface 190 may be a plain piece of paper, a specially formatteddocument, or any other type of writing surface.

In one embodiment, the upper portion of writing instrument 10 comprisestracking sensors 120, battery 130, transceiver 140 and a light emittingdiode (LED) or liquid crystal display (LCD) 150. In one embodiment,tracking sensors 120 comprise three accelerometers (122, 123 and 124)and three gyroscopes (126, 127 and 128); however, the number and type ofsensors may be varied depending on the accuracy desired and theenvironment in which writing instrument 10 is used. In one embodiment,tracking sensors 120 provide movement and acceleration in threedirections (e.g., x, y and z) with respect to the fixed frame of theposition tracking system.

Pen tip 180 and accelerometers 122, 123 and 124 are used to determinewhen the writing instrument is touching writing surface 190. In such anembodiment, only the strokes made when pen tip 180 is touching writingsurface 190 are processed. Of course, other embodiments are alsopossible, such as allowing a user to write in the air, with indicationsof which strokes are to be processed provided in another way, such asthe user pressing a button or activating a switch either on or offwriting instrument 10.

Battery 130 provides power to tracking sensors 120, transmitter/receiver140, display 150 and any other components of writing instrument 10 thatmay require power. Transceiver 140 transmits data from tracking sensors120 and other data to remote computing device 175. Transceiver 140 mayalso receive data from remote computing device 175. LED/LCD 150 providesfeedback to the user. Remote computing device 175 may be a personalcomputer, a fax machine, or other computing device.

To use writing instrument 10, a user writes a message on writing surface190. Pen tip 180 and ink supply 110 allow the user to write a message ashe or she normally would write on a piece of paper. As noted above,recording the message on writing surface 190 is not necessary; however,doing so provides the user with a copy of the message written for his orher records.

As the user writes out a message, tracking sensors 120 monitor thestrokes made to record the message written in the form of a series ofstrokes. Data from tracking sensors 120 are transmitted to remotecomputing device 175 via transceiver 140. Alternatively, the data fromtracking sensors 120 may be processed and stored in writing instrument10.

When a stroke is being made with writing instrument 10, tracking datafrom tracking sensors 120 are recorded and processed. In one embodiment,the stroke data output by the tracking sensors 120 are voltages. Foraccelerometers 122, 123 and 124, the voltages output are proportional tothe accelerations measured in each of three directions (e.g., x, y andz) with respect to the moving frame of writing instrument 10. Whenwriting instrument 10 is at rest, these accelerations consist of theforce of gravity (−9.81 m/sec²). For gyroscopes 126, 127 and 128, thevoltages output are proportional to the angular velocity of writinginstrument 10 in each of three directions (e.g., x, y and z) withrespect to the moving frame of writing instrument 10.

Voltage levels generated by the sensors (both accelerometers andgyroscopes) are transmitted to remote computing device 175 which mapsthe data to positions of writing instrument 10. Transmission of thevoltage levels may be accomplished, for example, by sampling the sensoroutput voltage levels with an analog-to-digital converter and sendingthe digital information to computing device 175 via transceiver 140.Computing device 175 receives the digital information representingvoltage levels output by the sensors and determines the position ofwriting instrument 10.

Prior to writing a message with writing instrument 10, calibration ofthe tracking system is performed by placing writing instrument 10 inknown positions taking sensor readings. The sensor readings obtained atthe known positions are used to define the known positions. Computingdevice 175 maps other sensor outputs to writing instrument positions byinterpolation based on the sensor outputs at the predeterminedpositions. Interpolation is performed by a polynomial approximation orother transform or neural network interpolation.

After calibration, voltage level information received by computingdevice 175 is mapped to positions of writing instrument 10. Bydetermining the position of writing instrument 10, computing device 175determines strokes made with writing instrument 10. Graphicalrepresentations of strokes are generated and output as a graphicaldisplay, a fax message, or for another purpose.

Alternative implementations of a position sensor system described asused in a writing instrument is disclosed are a patent applicationentitled “MANUAL ENTRY INTERACTIVE PAPER AND ELECTRONIC DOCUMENTHANDLING AND PROCESSING SYSTEM” filed Nov. 1, 1995, application Ser. No.08/551,535.

Position Tracking

In one embodiment, tracking of the position of writing instrument 10 isaccomplished via an Euler transform. The Euler transform provides fortransforming the output of tracking sensors 120 into data that definesstrokes made by the user with writing instrument 10. The Euler transformis a series of three frame rotations, starting with a ψ radian rotationabout the z-axis of a global (fixed) frame to produce frame 1. Thenframe 1 is rotated θ radians about its y-axis to produce frame 2 andfinally frame 2 is rotated φ radians about its x-axis to reach thewriting instrument frame. In one embodiment, frame rotations areperformed as described above; however, frame rotations may be performedin a different order depending, for instance, on which frame is desiredto have the greatest accuracy.

A Euler transform is used to transform acceleration and angular velocitydata into position data based on accelerations and velocities from aknown location. The Euler transform is commonly used in the fields ofaerial and naval navigation. When used for navigation, the Eulertransform is not calculated with the precision that is required fortracking of a writing instrument.

In order to recreate strokes made with writing instrument 10 from dataprovided by the Euler transform, accelerations are integrated twice todetermine distance relative to a starting position. Gyroscopes 126, 127and 128 in writing instrument 10 are used to correct for accelerationsinduced by angular rotations of writing instrument 10 as writinginstrument 10 is moved. As mentioned above, the computation required maybe performed by components of writing instrument 10 or by a remotedevice, such as remote computing device 175 in FIG. 1. Also,computations may be performed by hardware or software.

To convert accelerations in the writing instrument frame,

[{umlaut over (x)}_(w),{umlaut over (y)}_(w),{umlaut over (z)}_(w)],

into accelerations in the global frame,

[{umlaut over (x)}_(g),{umlaut over (y)}_(g),{umlaut over (z)}_(g)],

the following (Euler) transform is used: $\begin{matrix}{\begin{bmatrix}{\overset{¨}{x}}_{g} \\{\overset{¨}{y}}_{g} \\{\overset{¨}{z}}_{g}\end{bmatrix} = {{\begin{pmatrix}{\cos \quad {\theta cos}\quad \psi} & {{\sin \quad \varphi \quad \sin \quad \theta \quad \cos \quad \psi} - {\cos \quad \varphi \quad \sin \quad \psi}} & {{\cos \quad \varphi \quad \sin \quad \theta \quad \cos \quad \psi} + {\sin \quad \varphi \quad \sin \quad \psi}} \\{\cos \quad {\theta sin}\quad \psi} & {{\sin \quad \varphi \quad \sin \quad \theta \quad \sin \quad \psi} + {\cos \quad \varphi \quad \cos \quad \psi}} & {{\cos \quad \varphi \quad \sin \quad \theta \quad \sin \quad \psi} - {\sin \quad \varphi \quad \cos \quad \psi}} \\{{- \sin}\quad \theta} & {\sin \quad \varphi \quad \cos \quad \theta} & {\cos \quad \varphi \quad \cos \quad \theta}\end{pmatrix}\quad\begin{bmatrix}{\overset{¨}{x}}_{w} \\{\overset{¨}{y}}_{w} \\{\overset{¨}{z}}_{w}\end{bmatrix}}\quad {or}}} & {{Equation}\quad 1} \\{\begin{bmatrix}{\overset{¨}{x}}_{g} \\{\overset{¨}{y}}_{g} \\{\overset{¨}{z}}_{g}\end{bmatrix} = {{E^{- 1}\begin{bmatrix}{\overset{¨}{x}}_{w} \\{\overset{¨}{y}}_{w} \\{\overset{¨}{z}}_{w}\end{bmatrix}}.}} & {{Equation}\quad 2}\end{matrix}$

The dependence of the angles and sensor signals on time has been omittedto simplify the notation. The writing instrument frame is continuouslymoving, and sensor outputs are mapped into a global frame. Sensoroutputs are mapped at each time step. Time steps are chosen according tothe resolution desired, for example, 0.25 seconds.

Tracking of a writing instrument is more fully described in a U.S.patent application entitled “METHOD AND APPARATUS FOR TRANSFORMINGSENSOR SIGNALS INTO GRAPHICAL IMAGES”, application Ser. No. 08/996,537,filed Dec. 23, 1997, assigned to the corporate assignee of the presentinvention.

In addition to basic Euler position tracking, handwriting or patternrecognition may also be performed by writing instrument 10 or remotecomputing device 175. For example, the Euler transform parameters may beadjusted for known deviations from the true stroke. Also, other patternrecognition methods, such as Hidden Markov Modeling (HMM) or neuralnetworks, may be used.

Tracking System Calibration

Because the writing instrument may be used in many differentenvironments that can affect sensor output, such as temperaturevariations, altitude variations, etc., the present invention providestracking system calibration. Calibration may be accomplished by puttingthe sensors in known positions with known acceleration components, andthen using a polynomial approximation to interpolate between the(voltage, acceleration) pairs obtained at the known positions todetermine the position of the writing instrument. For example, when oneof the accelerometers, such as the x-axis accelerometer, isperpendicular to gravity, then the output of that accelerometer shouldcorrespond to −9.81 m/sec² and the other two accelerometers shouldcorrespond to 0 m/sec². When the writing instrument is rotated 90degrees, and the x-axis accelerometer is parallel to gravity, the outputshould correspond to 0 m/sec². The two points obtained (ν₁, −9.81) and(ν₂, 0) can then be fit to a voltage-position line so that new voltagesreceived represent a position of the writing instrument as determined bythe voltage-position line. Similar line fits are performed for the otheraccelerometers. If a higher order fit is desired, more readings may betaken and higher order equations, such as quadratic equations may beused to describe the voltage-position line.

In one embodiment, sensor readings comprise voltage levels proportionalto the acceleration measured by the accelerometers and to the angularvelocity measured by the gyroscopes. These voltage levels may be used,for example, as inputs to analog to digital converters, the outputs ofwhich are transmitted to a remote computing device and used to track theposition(s) of the writing instrument.

Because the surface that is perpendicular to the writing instrument maynot be perpendicular to gravity, two tilt angles (α,β) of that surfacemay be included in the transform used. In one embodiment, the user ofthe writing instrument holds the writing instrument in three predefinedpositions where each of the sensors is held perpendicular to the writingsurface. For each accelerometer output a slope and an intercept (m,b) isestimated to convert the voltage into an acceleration according to theequation:

(Acc=mV+b).  Equation 3

With these eight parameters (i.e., one slope and intercept for each ofthe tree accelerometers and two tilt angles) the following set ofequations is solved. Case 1: x-accelerometer perpendicular to thewriting surface:

g cos α cos β−{umlaut over ({overscore (x)})} _(p) m _(x) −b_(x)=0,  Equation 4

−g sin α−{umlaut over ({overscore (y)})} _(p) m _(y) −b _(y)=0,and  Equation 5

−g sin β−{umlaut over ({overscore (z)})} _(p) m _(z) −b_(z)=0.  Equation 6

Case 2: y-accelerometer perpendicular to the writing surface:

g sin α−{umlaut over ({overscore (x)})} _(p) m _(x) −b _(x)=0,  Equation7

g cos α cos β−{umlaut over ({overscore (y)})} _(p) m _(y) −b _(y)=0,and  Equation 8

−g sin β−{umlaut over ({overscore (z)})} _(p) m _(z) −b_(z)=0.  Equation 9

Case 3: z-accelerometer perpendicular to the writing surface:

g sin α−{umlaut over ({overscore (x)})} _(p) m _(x) −b _(x)=0,  Equation10

 g sin β−{umlaut over ({overscore (y)})} _(p) m _(y) −b _(y)=0,and  Equation 11

g cos α cos β−{umlaut over ({overscore (z)})} _(p) m _(z) −b_(z)=0.  Equation 12

A solution set for the eight parameters may be found by squaring each ofthe left hand sides of Equations 4-12, adding the results together, andthen reducing the resulting equation with respect to each parameter.This may be implemented in either hardware or software.

The solution set for the eight parameters discussed above provides amapping that allows sensor readings to be mapped to correspondingwriting instrument positions. The mapping provided by Equations 4-12 isa linear mapping. Higher order mappings may be accomplished with moresensor readings in additional predefined positions.

Physical Shape of the Writing Instrument

In order to simplify calibration for the user, writing instrument 10 mayhave one or more flat surfaces. These surfaces are held against thewriting surface to obtain calibration readings for the sensors. Afterthe readings are taken, the calibration calculations described above areperformed with the sensor readings. The result of the calibrationreadings are then used to calibrate the tracking system.

Alternatively, a stand or protractor-type device may be used to hold thewriting instrument in known positions. In such an implementation, thewriting instrument would not have to have flat surfaces for calibrationpurposes.

FIG. 2 is a perspective view of one embodiment of a writing instrumenthaving three calibrating surfaces. Writing instrument 10 has threecalibration surfaces 200, 210 and 230. The calibration sequence isdescribed below with respect to a particular sequence and angle betweencalibration surfaces; however, other angles and sequences may be used.In any case, the information that is known prior to a successfulcalibration sequence is the angular relationship between the calibrationsurfaces.

The user of writing instrument 10 first holds calibration surface 200against a writing surface. In the case of a writing surface that is apad of paper on a desk, the user simply places writing instrument 10 onthe pad of paper with calibration surface 200 down. Writing instrument10 is in this position for a period of time long enough to allow forstable sensor readings. In one embodiment, this period of time is 0.3seconds; however, other times may be used depending on the accuracy andspeed of the tracking system. If the writing surface is at an angle, thewriter may physically hold the writing instrument in place to preventwriting instrument 10 from sliding.

After writing instrument 10 has been placed on calibration surface 200,it is placed on calibration surface 210. In one embodiment, calibrationsurface 210 is at a right angle to calibration surface 200. Havingcalibration surfaces 200 and 210 at right angles allows the forces onwriting instrument 10 placed on a level surface to work only in onedirection. In other words, gravity is only measured along the x-axis,such that the y-axis and z-axis accelerometers sense no accelerations. Aright angle between calibration surfaces simplifies the calibrationcomputations.

The final calibration surface is calibration surface 230, which is thetop surface of writing instrument 10. In one embodiment, calibrationsurface 230 is at right angles to both calibration surface 200 andcalibration surface 210. After readings are taken with each calibrationsurface 200, 210 and 230 on the writing surface, the calibrationequations described above are evaluated and the output of the sensors inwriting instrument 10 are calibrated.

FIG. 3 is a perspective view of one embodiment of a writing instrumenthaving four calibrating surfaces. The embodiment of FIG. 3 provides ahigher order fit between voltage and accelerations because more readingsare taken. As with the embodiment of FIG. 2, the calibration sequence isdiscussed in a particular order and with a particular relationshipbetween the sides; however, other sequences and relationships may alsobe used.

Three of the four calibration surfaces are calibration surface 300,calibration surface 310 and calibration surface 320 connected to form atriangular upper portion of writing instrument 10. The fourthcalibration surface 330 forms the upper surface of writing instrument 10and is at a right angle to each of the other three calibration surfaces.Because the three calibration surfaces forming the triangular walls ofthe upper portion of writing instrument 10 are not at right angles toeach other the evaluation of the calibration equations above is morecomplex than in the case of right angles; however, more information isprovided and may be desired to improve the accuracy of the writinginstrument tracking. In the embodiment of FIG. 3, Equations 4-12 aremodified to compensate for the angles between the calibration surfacesbecause Equations 4-12 assume right angle relationships.

As is shown in the discussion above, any number of calibrating surfacesor positions may be used. What is required is that the relationshipbetween the positions is known. With this information and readings fromthe sensors, calibration of the sensors may be performed.

In the foregoing specification, the invention has been described withreference to specific embodiments thereof. It will, however, be evidentthat various modifications and changes may be made thereto withoutdeparting from the broader spirit and scope of the invention. Thespecification and drawings are, accordingly, to be regarded in anillustrative rather than a restrictive sense.

What is claimed is:
 1. A method comprising: obtaining sensor readingsfor each of a plurality of predefined positions of a writing instrumentgenerating a mapping of sensor outputs to writing instrument positionsbased, at least in part, on the sensor readings at the plurality ofpredefined positions with respect to a writing surface; and generating amapping of possible sensor outputs to possible writing instrumentpositions comprises determining a polynomial approximation of futurewriting instrument positions corresponding to future sensor outputsbased on sensor outputs at the predetermined positions, wherein thepolynomial approximation includes the sensor outputs for each predefinedposition.
 2. The method of claim 1 further comprising changing themapping of possible sensor outputs to possible writing instrumentpositions to compensate for an angle of a writing surface.
 3. The methodof claim 1 further comprising changing the mapping of possible sensoroutputs to possible writing instrument positions based on measurabledeviations of a mapped position to a known position.
 4. The method ofclaim 3, wherein changing the mapping of possible sensor outputs topossible writing instrument positions is based on pattern recognition.5. The method of claim 1, where in the sensors comprise a firstaccelerometer, a second accelerometer and a third accelerometer, whereinthe outputs of the accelerometers are voltages in proportion to theforce measured by the respective accelerometer.
 6. The method of claim1, wherein the sensors comprise a first gyroscope, a second gyroscope,and a third gyroscope, wherein the outputs of the gyroscopes arevoltages in proportion to angular velocities measured by the respectivegyroscopes.
 7. A method comprising: obtaining sensor readings for eachof a plurality of predefined positions of a writing instrument withrespect to a writing surface; and generating a mapping of sensor outputsto writing instrument positions based, at least in part, on the sensorreadings at the plurality of predefined positions and the elected sensorreadings at the plurality of predefined positions with respect to thewriting surface, wherein generating a mapping of possible sensor outputsto possible writing instrument positions comprises: mapping a firstvoltage output by a first sensor when the writing instrument is in afirst predetermined position with respect to the writing surface; andmapping a second voltage output by a second sensor when the writinginstrument is in a second predetermined position with respect to thewriting surface; and mapping sensor output voltages between the firstvoltage and the second voltage to positions of the writing instrumentbetween the first predetermined position and the second predeterminedposition.
 8. The method of claim 7, further comprising: mapping a thirdvoltage output by a third sensor when the writing instrument is in athird predetermined position with respect to the writing surface; andmapping sensor output voltages between the second voltage and the thirdvoltage to positions of the writing instrument between the secondpredetermined position and the third predetermined position.
 9. Themethod of claim 8, wherein the third predetermined position is at asecond right angle to the second predetermined position and at a thirdright angle to the first predetermined position.
 10. The method of claim8 wherein the third predetermined position a third accelerometer issubstantially perpendicular to gravity.
 11. The method of claim 7,wherein the first predetermined position is at a first right angle tothe second predetermined position.
 12. The method of claim 7 wherein thefirst predetermined position a first accelerometer is substantiallyperpendicular to gravity.
 13. The method of claim 12 wherein the secondpredetermined position a second accelerometer is substantiallyperpendicular to gravity.
 14. A system comprising: a plurality ofaccelerometers that generate output voltages in response toaccelerations in a corresponding direction; and a calibration circuit tomap the accelerometer output voltages to position data based onaccelerometer output voltages when the writing instrument is in aplurality predetermined positions and to generate a mapping of possiblesensor outputs to possible writing instrument positions comprisesdetermining a polynomial approximation of future writing instrumentpositions corresponding to future sensor outputs based on sensor outputsat the predetermined positions, wherein the polynomial approximationincludes the sensor outputs for each predefined position.
 15. The systemof claim 14, wherein the plurality of accelerometers comprise: a firstaccelerometer that outputs a first voltage when the writing instrumentis in a first predetermined position and a second voltage when thewriting instrument is in a second predetermined position; and a secondaccelerometer that outputs a third voltage when the writing instrumentis in the first predetermined position and a fourth voltage when thewriting instrument is in the second predetermined position; wherein thecalibration circuit maps the first voltage and the third voltage to thefirst predetermined position and maps the second voltage and the fourthvoltage to the second predetermined position, further wherein thecalibration circuit maps voltages between the first voltage and secondvoltage and voltages between the third voltage and the fourth voltage topositions of the writing instrument between the first predeterminedposition and the second predetermined position.
 16. The system of claim15 wherein the first predetermined position is at a first right anglewith respect to the second predetermined position.
 17. The system ofclaim 15 further comprising: a third accelerometer that outputs a fifthvoltage when the writing instrument is in the first predeterminedposition and a sixth voltage when the writing instrument is in thesecond predetermined position, and wherein the calibration circuit mapsthe fifth voltage to the first predetermined position and the sixthvoltage to the second predetermined position, and further wherein thecalibration circuit maps voltages between the fifth voltage and thesixth voltage to positions between the first predetermined position andthe second predetermined position.